ANALYSIS OF A NON-IDENTICAL COMPONENTSTRENGTHS SYSTEM BASED ON LOWER RECORD DATA Текст научной статьи по специальности «Математика»

ANALYSIS OF A NON-IDENTICAL COMPONENT-STRENGTHS SYSTEM BASED ON LOWER RECORD

DATA

Amal S. Hassan , Doaa M. Ismail, and Heba F. Nagy

Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, 12613, Egypt.

[email protected] [email protected] [email protected]

Abstract

In engineering applications and reliability literature, stress-strength models play a crucial role. The goal of this study is to develop more accurate stress-strength models by addressing the reliability estimation in multi-component systems with non-identical component strengths and stress. In the context of lower record values, the system's reliability is assessed using both classical and Bayesian approaches. In classical estimation, the maximum likelihood estimator of the reliability function is constructed, and a simulation study based on measurements of precision is used to assess the behavior of various estimates. The Bayesian estimators of reliability under general entropy, logarithmic and precautionary loss functions are computed. The suggested Bayesian estimates are calculated using the Markov Chain Monte Carlo method through a simulation study because there is no one particular way to do it. We found through simulated research that the accuracy of measurements decreases as the number of records rises. The theoretical results are validated using an example from actual data sets.

Keywords: Exponentiated Pareto model; Lower record data; Bayesian inference; General entropy loss function.

MSC: 62N05; 62D99; 62F15; 62F40; 94A20

1. Introduction

Record values are crucial when collecting observations is challenging or when they are lost during experimental operations. Real-world problems needing destructive stress testing, industrial quality control trials, and statistics on the weather, the economy, and sports all depend critically on record values. Only observations that exceed or fall below the most recent extreme values are recorded in this case. The total number of observations is frequently much lower than the overall sample size and only successive severe items are measured. Suppose that \Ui,i = 1,2,...} is an unlimited sequence of independent and identically distributed (iid). An observation Ui is called a lower record value (LRV) if Ui < U. for each i < j .

The stress-strength (SS) model is a fundamental of reliability testing. When a stress Y surpasses strength X, the SS framework ^ = P(X > Y) indicates the possibility of failure. In other words, the system keeps functioning as long as the stress does not outweigh its ability. Reference [1] was the author who initially presented this idea, and [2] later developed it. Many studies have

Amal S. Hassan, Doaa M. Ismail, and Heba F. Nagy RT&A, No 2 (73)

ANALYSIS OF A NON-IDENTICAL COMPONENT-STRENGTHS_Volume June 2023

addressed inferences based on various methods and distributional presumptions; for some recent works, see [3-10].

To build a system with two or more components, the basic idea of ' = P{X > Y) can be changed. Reliability in a multi-component SS (MSS) model was first created in [11], who investigated the MSS model under the assumption that c out of t system components, where, (1 < c < t ) components survive a common random stress Y. The MSS is applicable to several fields, including communications, logistics, military, and manufacturing operations. For illustration, if only four of a car's eight cylinders are burning, it might be possible to drive the vehicle. It can therefore be expressed as 4 out of 8: G system.

Assume that if c (1 < c < t) or more of the components cooperate, a system with t similar components will function. In its operational setting, the system is subjected to a stress Y that is a random variable with cumulative distribution function (cdf) G (y) . The component strengths, or the minimal stresses necessary to manufacture failure, are iid random variables with cdf F (x), then the reliability of the c-out-of-t system is represented by 'ct which is developed in [12], is given by:

'ct = P[atleastcof (X,X2,...,Xt) exceedY]

t (t\%

= £ . U[1 -F(x)]'F(x)'-'dG(x).

V—^ \ i I r-.

i=c

V

For various SS distributions and sampling procedures, many authors addressed the estimation of reliability in MSS models based on different sampling scheme, for example, see [1319].

Due to the different topologies of system components, the assumption of comparable strength distributions may not be feasible in many actual circumstances. With systems that have backup components, this is frequently the situation. The strengths of different items, even those constructed of the same material, can vary. For instance, heat treating metals to acquire desired mechanical properties in the field of mechanics can lead to various types of breaking when the metal is quenched or cooled. As a result, the strengths of the components vary. Another example, if two different ropes are used to consolidate a rope, the tensile strengths of both ropes may not be evenly distributed. A model that at least incorporates non-identical random strengths for system components appears to be more realistic, see [20].

Assume a system has t components, of which tj are of kind 1, t2 are of kind 2,..., and the

n—1

remainder tn = t — Xt, components are of kind n. Let F( ), i = 1, 2,..., n, be the cdf of the

random strengths for components of the ith kind. Assume that Y is a common stress with cdf Q Q that all components are subjected to. The system will function as long as the c-out-of-the-t components can resist the stress. Reference [21] presented the system reliability ^ c k t with

non-identical component strengths as follows:

ti tn ( " (t Z n

^...c*,...,, = X••• X n i I Ifn(1—F«y-(F(x)f-J'dQ(x), (1)

ji=ci jn=cn v -=1 v j )) 0 i=1

where summation ranges over all possible combinations (j,j2,...,j„) with 0< j <t for

n

i = 1, 2, ... , n such c <X ji < t. Each c indicates the minimal number of components of the

i=1

ith type that the system needs to function.

Considering the investigation of a system with two different sorts of components, the model (1) can be expressed as follows:

= I I [I It2 If(1 -^1(x))jl(^1(x))4-j1(1 -F2(x))2(F2(x))t2 jdQ(x). (2)

j =C j2 =c2 V j1 J V j2 J 0

In order to construct more realistic models, The Bayesian estimation of ^ k t assuming

the Weibull and exponential distributions on the strength and stress variates, respectively, was taken into consideration in [22] and [23]. The exponentiated Pareto distribution (EPD) was used to estimate № ^ t for non-identical MSS in [24]. Recently, [25] studied the case of non-identical component-strengths from the family of Kumaraswamy generalized distributions under upper record data. Reference [26] examined the estimation of № ^ t when component strengths and

stress follow inverse Lomax distribution based on complete sample. Reference [27] proposed the estimation of № ^ t when component strengths and stress follow Weibull distributions under

generalized progressive hybrid censoring scheme.

It is important to note that the majority of the work on the estimate of the SS reliability conducted to date requires to employ complete or censored samples and that record values are rarely used. Particularly in the estimation of MSS systems of non-identical component-strengths, we are interested in developing MSS models within the record scheme in the case of non-identical component-strengths, where component strengths and stress follow an EPD. A maximum likelihood estimator (MLE) of № is derived under LRV and a simulation study is

investigated. The general entropy loss function (GELF), the logarithmic loss function (LLF), and the precautionary loss function (PLF) are used to derive the Bayesian estimator of № t, t2. Since

these estimators are incapable of being reduced to simple closed forms, we use the Markov Chain Monte Carlo (MCMCO) approach for Bayesian estimates of № . To show the relevance of our work, we also examined real data sets.

The following is how the rest of the essay is presented. The formulation of № k t , and

its MLE under LRV along with a numerical analysis is provided in Section 2. Section 3 discusses the Bayesian estimators of № ^ t , through GELF, LLF, and PLF. The MCMCO technique is

presented in Section 4. For the purposes of illustration, Section 5 includes real data sets. Final remarks are included in Section 6.

2. Model Description and Classical Estimation of № C211

In this section, a model description of № is provided. The MLE of № is obtained in the presence of LRV. Numerical analysis is also carried out.

2.1. Expression of № A

Here, expression of W ti t is provided when the strength and stress random variables follow the EPD.

The EPD may be successfully used to assess numerous lifetime data sets, as argued in [28]. The EPD has a very flexible structure as a result of its decreasing or upside-down bathtub shape failure rates depending on shape parameters. This property provides advantages for modeling extreme events, particularly in hydrology. Furthermore, the EPD is a reasonable equivalent to the exponential distribution because of its heavier or lighter tail features. A variety of lifetime data might seem nice in the EPD. The EPD 's cdf is expressed by

s

F(x) =1 -(1 + x )-y , x >0, y>0,S>0,

where 5 and y are the shape parameters. The associated probability density function (pdf) is given by:

£-1

f (x) = Sy 1 -(1 + x )—y ' "(1 + x )—(y+1}, x > 0, y> 0,S> 0.

Several scholars addressed the EP's research and applications, for instance see [29]. From the total of t system components in the model (2), let the first t of first kind component strengths follow EPD (y, S), while the remaining t2 = t — t of kind 2 component strengths follow EPD (y,S2). Also, suppose that Y follows EPD (y,S3) independently. The respective distribution functions are as below:

F (x) = [l — (1 + x)—y J', x, y, St > 0, i = 1,2, (3)

r - ~\S3

Q(y) = [1 —(1 + y)—rJ , y,y,S3 >0. (4)

By replacing F, F, Q given in (2) by (3) and (4), the formula of ftCi C2 k t2 for such a system is as follows:

ft

C1 ,c2 ,'l ,'2

1 -2 Î1.Î2 =X X

Jl =C J2 =c2

'l V Jl 7

U

J (l - (l -(l + X )-r )ô) J1(1 -(l + x )-r )£1('1--J1)(1 - (l -(l + x yr )£2)J

x(l-(l + x)-r)s2(t2-J2)Sr[l-(l + x)-r]3- (l + x)-(r+l), Let z = l -(l + x)-r, dz = r(l + x )-r-1,

then ft„ , , is as follows

ft

'l '2 ft \t \ 1

^ W (1 - Z £l)Jl(1 - Z *)J2Z £('l -J2)+S3-1dz .

C1 ,c2 ,'l ,'2

J1=c1 J 2 =c2 V J1 7

VJ 2 y 0

Using the binomial expansion for (1 - zô )Jl and (1 - z02 )J2 leads to the following

ft , , = E. . ôA z

1

03 J z

ô (m+'l-Jl )+Ô2 (n+t2 -J2 )+ô3 -1,

Eô

jl,j2,m,n 3

(5)

ôl(m + tl - Jl) + Ô2(n +12 - J2) + Ô3

'l '2 Jl J2 ^ '

where E

Jl,J2,>

XXXX

Jl =Cl J2 =C2 m=0 n=0

.Jl.

^ v 7- v 7-

'2 If Jl

M.

m

Ji

(-1)"

Note that expression (5) depends on the parameters S, S and S3

2.2. The MLE of ft ,, under LRV

In order to find the MLE of ft based on LRV, we first need to obtain the MLE of the

C1 ,c2,t1,t2

parameters S, S2 and S3 assuming y is given known. So, let r ={r1,...,rn} p = {p1,...,pm } and S = {s,...,sw } be three independent sets of LRV of sizes n, m, and w from EPD (y,S), EPD (y,S2) and EPD (y, S3) respectively. The likelihood function of the observed records, according to [30], is defined by:

L(u\l) = h(ud )jQ -h(u'1

-<J<<... <u < J.

=1 H (ui ;r)

Hence, the observed LRV data r, p and s, given r, based on (6), are given as below:

(6)

0

Amal S. Hassan, Doaa M. Ismail, and Heba F. Nagy RT&A, No 2 (73) ANALYSIS OF A NON-IDENTICAL COMPONENT-STRENGTHS_Volume l8, June 2023

L( r, p, s \r) = r | y,S) L2( p | y,S2) L3( s | y,S3)

= — (y)n+m+w (1 + rn )—(y+1)(1 + pm )—(y+1)(1 + Sw )-(j+l)[Tn f1 —1

n-1

x[<m ]S2 ""X S "nd + r )-(y+1)(^, )-1 nd + Pj )-(y+1)(^j )-1 (7)

¿=1 j=1

w—1

x n(1 + )—(y+1)(^H)—1,

u=1

where rt = 1 —(1 + r)—y, <Pj = 1 — (1 + Pj)—y, and £ = 1 —(1 + U, i = 1,-,n, j = 1,...,m, u = 1,..., w.

Consequently, the joint log-likelihood function, denoted by In £, is derived as:

ln£ = win (dj + mln(d2) + wln(£3) +(n + m + w)ln(y) ~(y + l)ln(l + rn) + ln(l + pm) + ln(l +

n—1

+ (S — 1)ln[Tn ] + (S2 — 1)ln[Pm ] + (S3 — 1)ln[Cw ] — Z[(y + 1)ln(1 + r ) + )]

"3 V"1 LbwJ

1=1

—1

— X[(y + 1)ln(1 + Pj ) + ln(Pj )] — Z[(y + 1) ln(1 + S ) + ln(Cu )].

j=1 u=1

Given that ^ is a known, the following are the partial derivatives of In I with respect to ¿^,¿>2 and S3 respectively

fin' n , , , fin' ni , , fin' ii' . . „ . -= — + ln[r 1, -= — + lnfa> 1, -= — + lnfC,l.

/•»<■> <■> L n J' —so o L r m J' —so o L^wJ

oS S cS2 S S

Then, the MLEs of t>i,t>2 and<5, denoted by ¿>1, and <53 are obtained by setting rln ijc8], d\nt/dS2 and Sin i/853 to be zero. Hence Sl,82 and ¿ire obtained ¿is

—n —m ~ —w

S = S = —^S = ——. (8)

1 lnK ], 2 ln[Pm ], 3 ln[Cw ] ^ ( )

Therefore, based on invariance property, we obtain the MLE of " ^ t by inserting S, ¿>2 and S3 in (5) as follows

771 o

" cc.t =-,-j^-3-^. (9)

',2, ,2 S>1 (m +11 — j) + S,(n +12 — j2) + S,

2.3. Numerical Study

The MLE for the MSS variables is thoroughly numerically analysed in this subsection. In order to assess the accuracy of estimates for various parameter values and record numbers, two criteria are used: absolute biases (ABs) and mean squared errors (MSEs). The numerical research is performed in the following way:

Create LRV samples based on the parameter values provided.

The parameters values are selected as — = (0.5,0.5,0.2), (1.1,0.5,0.2), (0.5,0.5,0.5) and (1.1,1,0.2) for y = 1 in all situations. The specified values for c-out-of-t systems are (q, c2, t, t2 ) = (1,1,2,2), (1,2,2,2), (2,1,2,2) and (2,2,2,2).

The true values at (q, q, t, t2) = (1,1,2,2) are 0.533, 0.871, 0.758 and 0.809, at (q, c2, t, t2) = (1,2,2,2) are 0.3, 0.746,0.573 and 0.591, at (q, q, tx, t2) =(2,1,2,2) are 0.301, 0.760, 0.573 and 0.724, and at (q, q, t, h) = (2,2,2,2) are 0.2, 0.687, 0.477 and 0.562.

The sample sizes of LRV samples (n, m, w) are selected to be (2,2,2), (5,5,5), (7,7,7), (10,10,10), (2,2,3), (5,5,6), (7,7,8) and (10,10,11).

5000 repetitions are used to evaluate the ABs and MSEs of 5R h t .

The simulated outcomes are shown in Table 1 and are illustrated in Figures 1-6.

As the number of records increases, the MSEs of ftq c k t for all values of (c, c, t, h) decrease

(Figure 1). For all true value of parameters, the MSE of ftc c k ^decreases at (c, c, t, t2) = (2,2,2,2) when the number of records n = m (Figure 2).

0.01

0.005

m

0

Hi

<VV fP <V ^ ^ ^ ^ ^

(0.5,0.5,0.2) □ (1.1,0.5,0.2)

□ (0.5,0.5,0.5)

(1.1,1,0.2)

0.01

Si 0.005 m

3

m

m 01

□ (0.5,0.5,0.2)

□ (0.5,0.5,0.5)

(1.1,0.5,0.2) (1.1,1,0.2)

Figurel: MSEs of ftftc f Af for different (S1, S2, S3) values at (c,c2,tl,t2) = (1,1,2,2) and n = m = w

Figure 2: MSEs of Sft,^ for different (S1,S2,S3) values at (c, c2, ^ ,t2) = (2,2,2,2) and n=m

Figure 3 demonstrates that as the number of n, m and w increases, the ABs of ft k t for all actual values of (S1,S2,S?) are decreasing.

Figure 4 illustrates that the MSEs of ftq c k h at (c, c, A, h) = (2,2,2,2) are larger than the MSEs of ftc c tl ^ for others values of (c, c2, tx, t2).

Figure 3: ABs of ft for different (S1,S2,S3) values at (c, c2, ^,t2) = (1,2,2,2) and n = m = w

Figure 4: MSEs of ftc^,, for different (c^,^) values at n = m = w and (SuS2,S3) = (1.1,0.5,0.2)

Figure 5 illustrates that the MSEs of ftc c ^ i2 at (c, c2, t, h ) = (1,1, 2, 2) are smaller than the MSEs of ft h h for others values of (c, c2, t, t2 ) .

Figure 6 illustrates that the MSEs of ftCi C2 ^ t decrease when the true value of ft C2 k t increases.

0

*

0.01

§ 0.008 LU

<u 0.006

rs

m

0.004 0.002 0

JJJi

i-1-1-1-*i

(1,1,2,2) (1,2,2,2) (2,1,2,2) (2,2,2,2)

□ (0.5,0.5,0.2)

□ (0.5,0.5,0.5)

(1.1,0.5,0.2) (1.1,1,0.2)

0.015

§ 0.01 JJ 01

I 0.005

U" m c

7 0

R=0.562

R=0.591 R=0.724

R=0.809

jJjLL

^ ^ ^ ^

V V V V V 'Vi" V V

<v <V <V

Figure 5: MSEs of SHWl/! /or different (Su^S and Figure 6: MSEs of SHWl/! for (Su^S = d-l, 0.5, 0.2)

(q, q, tj ,t2) values at n = m = w =7 at n = m = w =2

Table1: Numerical results of c ,2 for different values of (-i,-2,-з)

(Si,S2,S3) = (0.5,0.5,0.2) (Si,S2,S3) = (1.1,0.5,0.2)

( c2, t1,t2 ) Real ( n, m, w) AB MSE ( ci, c2, ^2 ) Real ( n, m, w) AB MSE

(2,2,2) 0.0303 0.0009 (2,2,2) 0.0772 0.0060

(5,5,5) 0.0255 0.0006 (5,5,5) 0.0546 0.0029

(7,7,7) 0.0197 0.0003 (7,7,7) 0.0506 0.0025

(1,1,2,2) 0.758 (10,10,10) (2,2,3) 0.0182 0.0244 0.0002 0.0005 (1,1,2,2) 0.809 (10,10,10) (2,2,3) 0.0394 0.0284 0.0015 0.0008

(5,5,6) 0.0137 0.0002 (5,5,6) 0.0282 0.0007

(7,7,8) 0.0075 0.0001 (7,7,8) 0.0205 0.0004

(10,10,11) 0.0066 0.0001 (10,10,11) 0.0212 0.0004

(2,2,2) 0.0371 0.0013 (2,2,2) 0.0872 0.0076

(5,5,5) 0.0372 0.0013 (5,5,5) 0.0664 0.0044

(7,7,7) 0.0293 0.0008 (7,7,7) 0.0564 0.0031

(1,2,2,2) 0.573 (10,10,10) (2,2,3) 0.0233 0.0345 0.0004 0.0011 (1,2,2,2) 0.591 (10,10,10) (2,2,3) 0.0530 0.0628 0.0028 0.0039

(5,5,6) 0.0307 0.0009 (5,5,6) 0.0484 0.0023

(7,7,8) 0.0281 0.0007 (7,7,8) 0.0302 0.0009

(10,10,11) 0.0118 0.0001 (10,10,11) 0.0305 0.0009

(2,2,2) 0.0306 0.0009 (2,2,2) 0.0863 0.0074

(5,5,5) 0.0276 0.0007 (5,5,5) 0.0583 0.0034

(7,7,7) 0.0238 0.0005 (7,7,7) 0.0364 0.0026

(2,1,2,2) 0.573 (10,10,10) (2,2,3) 0.0210 0.0233 0.0003 0.0005 (2,1,2,2) 0.724 (10,10,10) (2,2,3) 0.0202 0.0303 0.0004 0.0009

(5,5,6) 0.0162 0.0003 (5,5,6) 0.0285 0.0008

(7,7,8) 0.0128 0.0002 (7,7,8) 0.0272 0.0007

(10,10,11) 0.0115 0.0001 (10,10,11) 0.0264 0.0007

(2,2,2) 0.0978 0.0095 (2,2,2) 0.1046 0.0109

(5,5,5) 0.0530 0.0028 (5,5,5) 0.0794 0.0063

(7,7,7) 0.0317 0.0010 (7,7,7) 0.0770 0.0059

(10,10,10) 0.0305 0.0008 (10,10,10) 0.0619 0.0038

(2,2,2,2) 0.477 (2,2,3) 0.0622 0.0038 (2,2,2,2) 0.562 (2,2,3) 0.0705 0.0049

(5,5,6) 0.0437 0.0019 (5,5,6) 0.0629 0.0039

(7,7,8) 0.0323 0.0010 (7,7,8) 0.0450 0.0020

(10,10,11) 0.0211 0.0001 (10,10,11) 0.0346 0.0012

(Si,52,S3> = (0.5,0.5,0.5) (Si,S2,S3) =(1.1,1,0.2)

(^ c2, t1,t2 ) Real ( n, m, w) AB MSE (^ c2, ti,t2 ) Real ft ( n, m, w) AB MSE

(2,2,2) 0.0972 0.0094 (2,2,2) 0.0267 0.0007

(5,5,5) 0.0958 0.0091 (5,5,5) 0.0222 0.0005

(7,7,7) 0.0922 0.0085 (7,7,7) 0.0190 0.0003

(1,1,2,2) 0.533 (10,10,10) (2,2,3) 0.0515 0.0707 0.0026 0.0050 (1,1,2,2) 0.871 (10,10,10) (2,2,3) 0.0183 0.0173 0.0001 0.0003

(5,5,6) 0.0628 0.0039 (5,5,6) 0.0130 0.0002

(7,7,8) 0.0395 0.0015 (7,7,8) 0.0122 0.0001

(10,10,11) 0.0251 0.0010 (10,10,11) 0.0113 0.0001

(2,2,2) 0.1002 0.0100 (2,2,2) 0.0423 0.0017

(5,5,5) 0.0984 0.0096 (5,5,5) 0.0296 0.0008

(7,7,7) 0.0977 0.0095 (7,7,7) 0.0217 0.0004

(1,2,2,2) 0.3 (10,10,10) (2,2,3) 0.0728 0.0717 0.0053 0.0049 (1,2,2,2) 0.746 (10,10,10) (2,2,3) 0.0176 0.0228 0.0002 0.0005

(5,5,6) 0.0684 0.0046 (5,5,6) 0.0216 0.0004

(7,7,8) 0.0558 0.0031 (7,7,8) 0.0128 0.0001

(10,10,11) 0.0301 0.0018 (10,10,11) 0.0121 0.0001

(2,2,2) 0.0957 0.0091 (2,2,2) 0.0396 0.0015

(5,5,5) 0.0954 0.0091 (5,5,5) 0.0288 0.0007

(7,7,7) 0.0947 0.0088 (7,7,7) 0.0215 0.0003

(2,1,2,2) 0.301 (10,10,10) (2,2,3) 0.0701 0.0720 0.0049 0.0051 (2,1,2,2) 0.760 (10,10,10) (2,2,3) 0.0170 0.0222 0.0001 0.0004

(5,5,6) 0.0636 0.0040 (5,5,6) 0.0211 0.0002

(7,7,8) 0.0422 0.0017 (7,7,8) 0.0121 0.0001

(10,10,11) 0.0288 0.0011 (10,10,11) 0.0118 0.0001

(2,2,2) 0.1028 0.0105 (2,2,2) 0.0430 0.0020

(5,5,5) 0.0530 0.0098 (5,5,5) 0.0317 0.0009

(7,7,7) 0.0317 0.0085 (7,7,7) 0.0220 0.0005

(2,2,2,2) 0.2 (10,10,10) (2,2,3) 0.0301 0.0906 0.0050 0.0082 (2,2,2,2) 0.687 (10,10,10) (2,2,3) 0.0178 0.0230 0.0003 0.0006

(5,5,6) 0.0437 0.0019 (5,5,6) 0.0235 0.0005

(7,7,8) 0.0323 0.0010 (7,7,8) 0.0130 0.0002

(10,10,11) 0.0299 0.0010 (10,10,11) 0.0128 0.0002

3. Bayesian Estimation of ft 11

We will look in this section at the Bayesian estimator of ft ^ t under the assumption that S ,S2 and S are random variables.

Following [31], the prior distributions for S ,S and S3 are assumed to have the gamma distribution with the following pdfs

7 (S ) ^ Sfl1 —^^, 7 (S ) ^ S°2—^^, and 73 (S ) ^ S°3 —e^,

where, the hyper-parameters; a1, ai, a3, hi, bi and b3 are considered to be known. The joint prior distribution of r = (SS,S2,S3), assuming parameters independence is as follows:

7c]r) = S fll—S °2—1S a—1leHb1S1+b2S2+b3S3).

Based on the observed samples, the joint density function of r = (S1,S2,S3) and the data are:

n(r \ L,p, s) = Sn+ °1—1Sm+ °2 —Sw+°3 —V(bS+bS+bS3)

\n+m+w(}j_v 1y+1)n^ n (y+1)n _1_S )—(y+1)[r ]S1—1

x (y)n+m+w (1 + L ) —(y+1)(1 + Pm ) —(y+1)(1 + Sw )—(y+1)[^n ]S

n—1 (1 . r ^— (y+1) m—1 (1 + p ) — (y+1) w—1 (1 N—(y+1)

4Pm S —1 [Cw S —1 n^— n^-pP<)- ^^^^

i=1 (Ti) j=1 (Pj ) u=1 (Cu)

As a result, the posterior density function of r = (S^^) can be expressed as

./ , \ L (L, P, S \ r)^(r)

n ( r \ L, p, S )=-----.

\ ' I — _ — J TO TO TO

HIL(L,p,S \ r)n(r)dSdS2dS3

000

The Bayesian estimator of v.)i ^ , based on GELF, indicated by V.K ^ (|is derived as follows

—1 CO CO CO

(10)

Additionally, the Bayesian estimator of 9t C2 k h , under LLF indicated by 91 C2 is as follows:

q2,t1,t2

TO TO TO

= exp(£(log9iCi C2 i2) = exp

|||log,„2n (r \ L, AS)d■—ld■—2dS3

. (11)

The Bayesian estimator of v.)i ^ (| ^ for PLF indicated by k h is as follows:

__CO CO CO

"1,^1,, =4Eq1,q2,t1,t2) = III«2d^^^n* (rl L,p,s)dS1dS2dS3

(12)

It is difficult to find an explicit formula for (10)-(12) because the posterior density function n (r \ L, p, S) has a composite structure. In order to obtain Bayesian estimates, we calculate these integrations using the Metropolis-Hastings (M-H) technique using the MCMCO algorithm.

4. MCMCO Methodology

The MCMCO simulation is used to investigate the behavior " c k t 's MSS. Bayes estimates (BE) under different loss functions are produced using gamma priors. The " c h t 's BE accuracy is measured using the ABs, and MSEs. The various LRV options are (n, m, w) = (2, 2, 2), (5, 5, 5), (7, 7, 7), (10, 10, 10), (2, 2, 3), (5, 5, 6), and (7, 7, 8). The possible sets of hyperparameter values are considered to be: Prior I: (2, 1.5, 3, 2, 1.5, 1.1) and Prior II:(1, 1.4, 1, 2, 2.5, 3).

The outcomes are based on 5,000 replications. The M-H process is a popular subgroup of the MCMCO technique in the Bayesian literature for modeling departures from the posterior density and producing accurate anticipated results. The main difficulty with the MCMCO is getting the BEs of « from GELF, LLF, and PLF using the M-H approach after simulating

samples from the posterior density. It converges to the desired distribution using acceptance/rejection criteria. The M-H algorithm (see [32]) operates as follows:

a) Set the starting parameter value of C2 k, and the sample number N.

b) For i = 2 to N, set q 11 = 1q q 11 .

c) Create u using the uniform (0,1).

d) Choose a candidate parameter from the proposal density.

0.5

7(0*) g (o0) . .

e) If u < y ' ; , / , then set ft' , = ft , , , , ; otherwise, set ft'

' zs/lilsV c1 ,c2 ,t1 ,t2 c ,c2 ,t1 ,t2

c1,c2,t1,t2

= ft„

c1 ,c2 ,t1 ,t2

7(0) g (o|o*)

f) Return to step (b) and perform the aforementioned actions N times using t = t+1.

Using the outputs of the study, which are shown in Tables 2, 3, and are illustrated by Figures 7-12, we come up with the following conclusions:

• The MSEs and ABs of ftc c ti t estimates via the GELF, LLF and PLF decrease with increasing the record numbers n, m, w rises for all true values of (c, c2, t, t2), (Tables 2, 3).

• The ABs of ft estimates via the GELF, LLF and PLF have the smallest values at

c1,c2,t1,t2

(cl5 c2, tx, t2) = (1,1, 2, 2), (Tables 2, 3).

• At true value S.H c k, = 0.748, the MSE of ft c (| ( via PLF take the smallest values in case of prior I except at (7, 7, 8) (see Figure 7).

• At true value ft , , =0.748, the AB of ft . . , , at PLF gets the fewest values for a distinct

c1 ,c2 ,*1,t2 c1,c2 ,t1 ,t2 '

number of records excepting at (n, m, w) = (7,7,8) via prior I (see Figure 8).

T> ^ ^ ^

O* A' O.? <5' A'

<V <V ^ <V <V

ft.

,9i

Figure 7: MSEs of ft^^ , --c^Wc^aj, (c,c2t) = (1, 2, 2, 2) for prior I

at

Figure 8: ABs of ft

ft

□ PLF

— t t ^ t t

at

(c, c ,t2) = (1,1, 2, 2) for prior I

The MSEs of ft c (| ( , ft c (| h ,91 c (| ( under the GELF, LLF and PLF, respectively, decease as the number of records n = m = 10 increases via prior II (see Figure 9).

At true value = 0.760, the MSEs of ft^,^, under the GELF, LLF

and PLF, respectively, get the least values for similar record values of (n, m) via prior II (see Figure 10).

2 >1 >2

0.0014 o 0.0012 0.001

% 0.0006

0.0004 0.0002 0

GELF

LLF □ PLF

Figure 9: MSEs of «'"' «'

(q,q,ti,t2) = (2 ,l, 2, 2) for prior II

(2,2,2) (5,5,5) (2,2,3) (5,5,7)

GELF LLF PLF

Figure 10: MSEs of and \ at

true value c h = 0.760 for prior II

At true value 4 u =0.746, the MSEs of 9R ^ t h gets the smallest values compared to 9? c (|, , and 9? c (|, for similar record values except at («, w) = (10,10,10) via prior II (see Figure 11).

Figure 12 illustrates that the ABs of 9? c (|

9? c t ( ,9? c t ( decease as true value of

q1 ,q2 ,t1 ,t2

increases for (n, m, w) = (2,2,2).

0.0006

^

<>>' # GELF LLF PLF

Figure 11: MSEs of and \ at

(q, q ,tj ,t2) =(l, 2, 2, 2) for prior II

R=0.687

^ ^ ^ ^

V "V V V V V V V <V <V

GELF LLF PLF

Figure 12: The ABs for all true values of « at n = m = w = 2 for prior II

Table 2: Numerical results of 9t ^ ^ ^ , 91 ^ ^ u , 91 ^ ^ ^ for pr/or I

(ci, c2, ti, t2) = (1,2,2,2) (C1, C2, t1, t2) =(1,2,2,2)

Loss function Real (n, m, w) Real (n, m, w)

AB MSE AB MSE

C1 ,C2 ,?2 C1 ,C2 ,?2

GELF 0.871 0.02036 0.00041 0.746 0.04300 0.00184

LLF (2, 2, 2) 0.01541 0.00023 (2, 2, 2) 0.03373 0.00113

PLF 0.006413 0.00004 0.01229 0.00015

GELF 0.00542 2.9E-05 0.04175 0.00174

LLF (5,5,5) 0.00048 2.3E-07 (5,5,5) 0.03295 0.00108

PLF 0.00067 4.5E-07 0.01169 0.00013

GELF (7,7,7) 0.00314 9.9E-06 (7,7,7) 0.03750 0.00140

LLF 0.00036 1.3E-07 0.02955 0.00087

PLF 0.00023 5.4E-08 0.01078 0.00011

GELF 0.00293 8.6E-06 0.00323 1.0E-05

LLF (10,10,10) 0.00030 1.3E-07 (10,10,10) 0.00289 8.3E-06

PLF 0.00020 4.2E-08 0.00005 3.2E-09

GELF 0.02601 0.00067 0.03778 0.00142

LLF (2,2,3) 0.02085 0.00043 (2,2,3) 0.02893 0.00083

PLF 0.00891 0.00007 0.01107 0.00012

GELF 0.01449 0.00021 0.02832 0.00080

LLF (5,5,7) 0.00995 0.00009 (5,5,7) 0.02031 0.00041

PLF 0.00407 1.6E-05 0.00797 6.3E-05

GELF LLF PLF

(7,7,8)

0.000286 0.00103 0.00058

8.1E-06 1.0E-06 3.3E-07

(7,7,8)

0.00768 0.00161 0.00177

5.9E-05 2.6E-06 3.1E-06

(ci, c2, ti, t2) = (2,1,2,2)

(ci, c2, ti, t2) = (1,2,2,2)

Loss function

Real

ft

(n, m, w)

AB

MSE

ft

Real

Cj ,C2 ,ti ,t2

(n, m, w)

AB

MSE

GELF LLF PLF

0.760

(2,2,2)

0.04292 0.03398 0.01359

0.00184 0.00115 0.00018

0.687

(2,2,2)

0.02976 0.019006 0.00830

0.00088 0.00036 6.8E-05

GELF 0.03351 0.00112 0.01627 0.00026

LLF (5,5,5) 0.02530 0.00064 (5,5,5) 0.00513 2.6E-05

PLF 0.00979 9.5E-05 0.00013 1.8E-08

GELF 0.02497 0.00062 0.00552 3.0E-05

LLF (7,7,7) 0.01697 0.00028 (7,7,7) 0.00244 5.9E-06

PLF 0.00664 4.4E-05 0.00016 2.6E-08

GELF 0.01358 0.00051 0.00491 6.11E-10

LLF (10,10,10) 0.01511 0.00017 (10,10,10) 0.00235 2.04E-11

PLF 0.00544 3.2E-05 0.00015 2.03E-10

GELF 0.04676 0.00218 0.02680 0.01649

LLF (2,2,3) 0.04676 0.00144 (2,2,3) 0.01649 0.00027

PLF 0.01530 0.00023 0.00663 4.41E-05

GELF 0.02963 0.00087 0.00534 2.8E-05

LLF (5,5,7) 0.02181 0.00047 (5,5,7) 0.00193 3.7E-06

PLF 0.01002 0.00010 0.00187 3.5E-06

GELF 0.00357 1.2E-05 0.00017 3.11E-10

LLF (7,7,8) 0.00270 7.3E-06 (7,7,8) 0.00105 6.55E-11

PLF 0.00031 9.9E-08 0.00135 3.44E-10

Table 3: Numerical results of 91

(C1,C2, ti, t2) = (1,1,2,2)

(ci, C2, t1, t2) = (1,2,2,2)

function ftr r t t Ci ,¿2 (n, m, w) AB MSE ft (n,m,w) AB MSE

Cj ,C2 ,ti t

GELF 0.871 0.01134 0.00012 0.746 0.02343 0.00054

LLF (2,2,2) 0.00566 3.2E-05 (2,2,2) 0.01414 0.00020

PLF 0.002903 8.4E-06 0.005499 3.0E-05

GELF 0.00355 1.2E-05 0.02311 0.00047

LLF (5,5,5) 0.00161 2.6E-06 (5,5,5) 0.01241 0.00015

PLF 0.00072 5.2E-07 0.00421 2.7E-05

GELF 0.00206 4.2E-06 0.02277 0.00037

LLF (7,7,7) 0.00153 2.3E-06 (7,7,7) 0.01187 0.00011

PLF 0.00103 1.8E-07 0.00365 1.1E-04

GELF LLF (10,10,10) 0.00201 0.00140 3.4E-06 1.2E-07 (10,10,10) 0.02148 0.01099 0.00029 3.0E-05

PLF 0.00099 5.2E-08 0.00301 1.0E-04

GELF 0.02524 0.00164 0.03074 0.00426

LLF (2,2,3) 0.02358 0.00121 (2,2,3) 0.03009 0.00077

PLF 0.00799 0.00080 0.00784 1.7E-05

GELF 0.02470 0.00135 0.02457 0.00333

LLF (5,5,7) 0.02157 0.00117 (5,5,7) 0.02847 0.00051

PLF 0.00630 0.00060 0.00780 1.1E-05

GELF 0.02110 0.00124 0.01354 0.00251

LLF (7,7,8) 0.01110 0.00101 (7,7,8) 0.02147 0.00039

PLF 0.00558 0.00038 0.00660 1.8E-06

(cx, c2, tx, t2) = (2,1,2,2) (C1,C2, tx, t2) = (2,2,2,2)

Loss function Real M , , Real M , ,

( n, m, w) AB MSE ( n, m, w) AB MSE

Cx 9C2 ,'2

GELF 0.760 0.02320 0.00136 0.687 0.02377 0.00056

LLF (2,2,2) 0.01312 0.00078 (2,2,2) 0.01288 0.00016

PLF 0.00445 0.00008 0.00515 2.6E-05

GELF 0.03524 0.00128 0.02228 0.00035

LLF (5,5,5) 0.02600 0.00071 (5,5,5) 0.01147 0.00012

PLF 0.00931 8.1E-05 0.00478 1.8E-06

GELF 0.03421 0.00088 0.02147 2.4E-05

LLF (7,7,7) 0.02387 0.00050 (7,7,7) 0.01133 2.9E-06

PLF 0.00900 2.4E-05 0.00330 2.9E-08

GELF 0.03321 0.00051 0.02140 6.1E-08

LLF (10,10,10) 0.02340 0.00044 (10,10,10) 0.01110 2.0E-07

PLF 0.00875 3.2E-06 0.00250 2.0E-08

GELF 0.03476 0.02336 0.02131 0.01356

LLF (2,2,3) 0.03554 0.00744 (2,2,3) 0.01109 0.00124

PLF 0.02447 0.00037 0.00190 2.4E-05

GELF 0.03124 0.01235 0.02110 0.01254

LLF (5,5,7) 0.02490 0.00625 (5,5,7) 0.01148 0.00120

PLF 0.02300 0.00035 0.00166 1.4E-05

GELF 0.03009 0.00147 0.01999 0.00124

LLF (7,7,8) 0.02370 0.00420 (7,7,8) 0.01122 0.00110

PLF 0.02298 0.00021 0.00150 1.0E-06

Note that: E-0k stands for 10-k, k is integer

5. Actual Data Implementation

In this part, we illustrate our principles using three real datasets. We consider the real data sets reported in [33] where the data represent the time to break down (in minutes) of insulating fluids to electrodes at voltage levels 34 kV, 36 kV and 38 kV. The Kolmogorov-Smirnov (KS) test is used to separately fit each of the three datasets with the EPD along with the corresponding P-value (PV) (see Table 4). The empirical cdf and estimated pdf for these data are explained in Figure 13. At levels 34 kV, 36 kV and 38 kV, the times to break down are reported respectively as follows Data Group I

31.75 7.35 6.5 8.27 33.91 1.31 12.06 36.71 72.89

0.19 4.85

0.78 2.78

8.01 4.67

0.96 4.15 32.52 3.16 Data Group II

1.97 0.59 2.58 1.69 2.71 25.5 2.07 0.96 5.35 2.9 13.77

Data Group III

0.47 0.73 1.4 0.74 0.39

0.35

1.13

0.99

3.9

3.67

0.09

2.38

Table 4: The K-S test and corresponding P-values for groups I, II and III

Data K-S PV

Group I 0.167 0.6013

Group II 0.185 0.6127

Group III 0.277 0.5013

Data I

Data II

Data III

Figure 13: Characteristics and limitations of K-S test for the three data groups

We assume that electrical fluid of specimen considered being good if 1 out of 2 specimens are functioning properly at constant voltage. Form data group I, II and III, three sets of lower record values r = (0.96, 0.19), p = (1.97, 0.59, 0.35) and s = (0.47, 0.39, 0.09) are obtained, respectively. From

r, p, and s, we find that n = 2, m = 3, w = 3, then we calculate the estimates of ft^c k t2 using

the ML and Bayesian approaches within GELF, LLF and PLF. Using the above LRVs, the MLE and

BE of ft , are calculated in Table 5.

c1,c2,t1,t2

Table 5: Bayes and ML estimates of ftc c h ,2, for the real data

BE of c t t c1 >c2 '*1 >*2

MLE of c , , c1 >c2 '*1 > 2

GELF LLF PLF

0.5851 0.7673 0.7743 0.7956

Amal S. Hassan, Doaa M. Ismail, and Heba F. Nagy RT&A, No 2 (73) ANALYSIS OF A NON-IDENTICAL COMPONENT-STRENGTHS_Volume 18, June 2023

6. Concluding Remarks

In the present work, we investigate the stress-strength reliability in a multi-component system with non-identical component strengths where both the stress and strength variables are the EPD. The ML and Bayesian procedures are used to analyse the reliability of MSS. Strength and stress distribution samples are used, and their measurements are presented in LRVs. We use MCMCO techniques in order to evaluate the accuracy of the various Bayesian estimates. The simulation study shows that for four choices of (c, c, t, h ), the MSEs and ABs decrease with the number of records, supporting the MLE's consistency characteristic of ft . Additionally, as the true

value of "ft c k h increases, the MSEs of ft h t drop. Regarding the MCMCO approach, we

deduce that the MSEs and ABs of 91 ^ (| h via PLF generally hold the lowest values in majority of

cases. The ABs and MSEs of ^ (| t, ^ (| t ,31 ^ (| t under different loss functions decrease as

the number of records rises. The use of actual data demonstrates that our model's reliability estimates are very near to one, demonstrating its practical usefulness.

Conflicts of Interest: The authors declare no conflict of interest.

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